There’s quite a bit of discussion in this thread and many others on the subject of whether or not a Autopilot will/should follow vertical guidance when on an RNAV approach if you use the search command.
Essentially, RNAV approaches are lateral only approaches, but, it’s not quite as simple as that as far as autopilots and GPS are concerned.
Here’s an excellent thread on the subject
I personally can’t answer your question. I’m not IFR rated (I do have my license), and I’m not familiar with the equipment across the various planes. It really is equipment dependent on the capabilities of the autopilot and it’s ability to communicate with the GPS.
But as much as I’ve studied, it doesn’t and it shouldn’t. But, don’t hold me to that. Do some research. That’s what I’ve figured out from my own research. And I could very well be wrong.
As I understand it, the pilot is meant to hand fly the altitude down to the decision point, and let the autopilot control the direction. But, I think it’s really dependent on the autopilot on what the answer is.
Somebody else posted this video, and said it’s proof it should. But I watched it and it seemed to me like it was proof it shouldn’t. It’s an example of hand flying an RNAV approach, so, it’s not directly applicable.